In this paper, we provide a new effective method for computing the exact value of Hausdorff measures of a class of self-similar sets satisfying the open set condition (OSC). As applications, we discuss a self-simila...In this paper, we provide a new effective method for computing the exact value of Hausdorff measures of a class of self-similar sets satisfying the open set condition (OSC). As applications, we discuss a self-similar Cantor set satisfying OSC and give a simple method for computing its exact Hausdorff measure.展开更多
We analyze the local behavior of the Hausdorff centered measure for self- similar sets. If E is a self-similar set satisfying the open set condition, thenC^s(E∩B(x,r))≤(2r)^sfor all x ∈ E and r〉 0, where Cs ...We analyze the local behavior of the Hausdorff centered measure for self- similar sets. If E is a self-similar set satisfying the open set condition, thenC^s(E∩B(x,r))≤(2r)^sfor all x ∈ E and r〉 0, where Cs denotes the s-dimensional Hausdorff centered measure. The above inequality is used to obtain the upper bound of the Hausdorff centered measure. As the applications of above inequality, We obtained the upper bound of the Hausdorff centered measure for some self-similar sets with Hausdorff dimension equal to 1, and prove that the upper bound reach the exact Hausdorff centered measure.展开更多
Let S belong to R^2 be the attractor of the iterated function system {f1, f2, f3 } iterating on the unit equilateral triangle So. where fi(x) =λix + bi, i = 1,2, 3, x =(x1, x2), b1=(0, 0), b3=(1-λ3 /2,√3...Let S belong to R^2 be the attractor of the iterated function system {f1, f2, f3 } iterating on the unit equilateral triangle So. where fi(x) =λix + bi, i = 1,2, 3, x =(x1, x2), b1=(0, 0), b3=(1-λ3 /2,√3/2 (1-λ3)) This paper determines the exact Hausdorff measure, centred covering measure and packing measure of S under some conditions relating to the contraction parameter.展开更多
It is important to calculate the Hausdorff dimension and the Hausdorff mesure respect to this dimension for some fractal sets. By using the usual method of “Mass Distribution”, we can only calculate the Hausdorff di...It is important to calculate the Hausdorff dimension and the Hausdorff mesure respect to this dimension for some fractal sets. By using the usual method of “Mass Distribution”, we can only calculate the Hausdorff dimension. In this paper, we will construct an integral formula by using lower inverse s-density and then use it to calculate the Hausdorff measures for some fractional dimensional sets.展开更多
The self similar sets satisfying the open condition have been studied. An estimation of fractal, by the definition can only give the upper limit of its Hausdorff measure. So to judge if such an upper limit is its exac...The self similar sets satisfying the open condition have been studied. An estimation of fractal, by the definition can only give the upper limit of its Hausdorff measure. So to judge if such an upper limit is its exact value or not is important. A negative criterion has been given. As a consequence, the Marion’s conjecture on the Hausdorff meas\| ure of the Koch curve has been proved invalid.展开更多
By means of the idea of [2](Jia Baoguo,J.Math.Anal.Appl.In press) and the self.similarity of Sierpinski carpet, we obtain the lower and upper bounds of the Hausdorff Measure of Sierpinski carpet, which can approach ...By means of the idea of [2](Jia Baoguo,J.Math.Anal.Appl.In press) and the self.similarity of Sierpinski carpet, we obtain the lower and upper bounds of the Hausdorff Measure of Sierpinski carpet, which can approach the Hausdorff Measure of Sierpinski carpet infinitely.展开更多
First of all the authors introduce the concepts of random sub-self-similar set and random shift set and then construct the random sub-self-similar set by a random shift set and a collection of statistical contraction ...First of all the authors introduce the concepts of random sub-self-similar set and random shift set and then construct the random sub-self-similar set by a random shift set and a collection of statistical contraction operators.展开更多
In this article, the Hausdorff dimension and exact Hausdorff measure function of any random sub-self-similar set are obtained under some reasonable conditions. Several examples are given at the end.
A set in Rd is called regular if its Hausdorff dimension coincides with its upper box counting dimension. It is proved that a random graph-directed self-similar set is regular a.e..
The structure of any a.s. self-similar set K(w) generated by a class of random elements {gn,wσ} taking values in the space of contractive operators is given and the approximation of K(w) by the fixed points {Pn,wσ} ...The structure of any a.s. self-similar set K(w) generated by a class of random elements {gn,wσ} taking values in the space of contractive operators is given and the approximation of K(w) by the fixed points {Pn,wσ} of {gn,ow} is obtained. It is useful to generate the fractal in computer.展开更多
We constructed a class of generalized statistically self-similar set.S and give the necessary and sufficent conditions to ensure a random recursive set being a generalized statistically self-similar set. The statist...We constructed a class of generalized statistically self-similar set.S and give the necessary and sufficent conditions to ensure a random recursive set being a generalized statistically self-similar set. The statistically self-similar sets defined by Hutchinson,Falconer,Graf are the special cases of ours.展开更多
In this paper, some results on the upper convex densities of self-similar sets at the contracting-similarity fixed points are discussed. Firstly, a characterization of the upper convex densities of self-similar sets a...In this paper, some results on the upper convex densities of self-similar sets at the contracting-similarity fixed points are discussed. Firstly, a characterization of the upper convex densities of self-similar sets at the contracting-similarity fixed points is given. Next, under the strong separation open set condition, the existence of the best shape for the upper convex densities of self-similar sets at the contracting-similarity fixed points is proven. As consequences, an open problem and a conjecture, which were posed by Zhou and Xu, are answered.展开更多
In this paper, authors compute the Packing dimension of statistically selfsimilar sets and obtaine the dimension and dimension distribution of statistically self-similar measure.
In this paper, we present a more simple and much shorter proof for the main result in the paper " An negative answer to a conjecture on the self-similar sets satisfy- ing the open set condition", which was published...In this paper, we present a more simple and much shorter proof for the main result in the paper " An negative answer to a conjecture on the self-similar sets satisfy- ing the open set condition", which was published in the journal Analysis in Theory and Applications in 2009.展开更多
Falconer[1] used the relationship between upper convex density and upper spherical density to obtain elementary density bounds for s-sets at H S-almost all points of the sets. In this paper, following Falconer[1], we ...Falconer[1] used the relationship between upper convex density and upper spherical density to obtain elementary density bounds for s-sets at H S-almost all points of the sets. In this paper, following Falconer[1], we first provide a basic method to estimate the lower bounds of these two classes of set densities for the self-similar s-sets satisfying the open set condition (OSC), and then obtain elementary density bounds for such fractals at all of their points. In addition, we apply the main results to the famous classical fractals and get some new density bounds.展开更多
Let d ≥ 1 be an integer and E a self-similar fractal set, which is the attractor of a uniform contracting iterated function system (UIFS) on R^d. Denote by D the Hausdorff dimension, by H^D(E) the Hausdorff measu...Let d ≥ 1 be an integer and E a self-similar fractal set, which is the attractor of a uniform contracting iterated function system (UIFS) on R^d. Denote by D the Hausdorff dimension, by H^D(E) the Hausdorff measure and by diam(E) the diameter of E. If the UIFS is parametrised by its contracting factor c, while the set ω of fixed points of the UIFS does not depend on c, we will show the existence of a positive constant depending only on ω, such that the Hausdorff dimension is smaller than one and H^O(E) = diam(E)^D if c is smaller than this constant. We apply our result to modified versions of various classical fractals. Moreover, we present a parametrised UIFS, where ω depends on c and show the inequatily H^D(E) 〈 diam(E)^D, if c is small enough.展开更多
We study the Hausdorff measure of linear Cantor setE, on the unit interval, under the strong seperated condition. We give a necessary and sufficient condition for ?(E)=∣E∣° by using the contracting ratio and th...We study the Hausdorff measure of linear Cantor setE, on the unit interval, under the strong seperated condition. We give a necessary and sufficient condition for ?(E)=∣E∣° by using the contracting ratio and the first gap. This condition is easy to use. Key words linear Cantor set - Hausdorff measure - strong seperated condition CLC number O 174. 12 Foundation item: Supported by the National Natural Science Foundation of China (10171028)Biography: Ma Chao (1975-), male, Ph. D. candidate, research direction: fractal geometry.展开更多
基金Supported in part by Education Ministry, Anhui province, China (No. KJ2008A028)
文摘In this paper, we provide a new effective method for computing the exact value of Hausdorff measures of a class of self-similar sets satisfying the open set condition (OSC). As applications, we discuss a self-similar Cantor set satisfying OSC and give a simple method for computing its exact Hausdorff measure.
基金supported by the National Natural Science Foundation of China (No. 11371379)
文摘We analyze the local behavior of the Hausdorff centered measure for self- similar sets. If E is a self-similar set satisfying the open set condition, thenC^s(E∩B(x,r))≤(2r)^sfor all x ∈ E and r〉 0, where Cs denotes the s-dimensional Hausdorff centered measure. The above inequality is used to obtain the upper bound of the Hausdorff centered measure. As the applications of above inequality, We obtained the upper bound of the Hausdorff centered measure for some self-similar sets with Hausdorff dimension equal to 1, and prove that the upper bound reach the exact Hausdorff centered measure.
基金the Foundation of National Natural Science Committee of Chinathe Foundation of the Natural Science of Guangdong Provincethe Foundation of the Advanced Research Center of zhongshan University
文摘Let S belong to R^2 be the attractor of the iterated function system {f1, f2, f3 } iterating on the unit equilateral triangle So. where fi(x) =λix + bi, i = 1,2, 3, x =(x1, x2), b1=(0, 0), b3=(1-λ3 /2,√3/2 (1-λ3)) This paper determines the exact Hausdorff measure, centred covering measure and packing measure of S under some conditions relating to the contraction parameter.
文摘It is important to calculate the Hausdorff dimension and the Hausdorff mesure respect to this dimension for some fractal sets. By using the usual method of “Mass Distribution”, we can only calculate the Hausdorff dimension. In this paper, we will construct an integral formula by using lower inverse s-density and then use it to calculate the Hausdorff measures for some fractional dimensional sets.
文摘The self similar sets satisfying the open condition have been studied. An estimation of fractal, by the definition can only give the upper limit of its Hausdorff measure. So to judge if such an upper limit is its exact value or not is important. A negative criterion has been given. As a consequence, the Marion’s conjecture on the Hausdorff meas\| ure of the Koch curve has been proved invalid.
文摘By means of the idea of [2](Jia Baoguo,J.Math.Anal.Appl.In press) and the self.similarity of Sierpinski carpet, we obtain the lower and upper bounds of the Hausdorff Measure of Sierpinski carpet, which can approach the Hausdorff Measure of Sierpinski carpet infinitely.
基金Supported by the National Natural Science Foundation of China (10371092)the Foundation of Wuhan University
文摘First of all the authors introduce the concepts of random sub-self-similar set and random shift set and then construct the random sub-self-similar set by a random shift set and a collection of statistical contraction operators.
基金supported by the National Natural Science Foundation of China(10371092)Foundation of Ningbo University(8Y0600036).
文摘In this article, the Hausdorff dimension and exact Hausdorff measure function of any random sub-self-similar set are obtained under some reasonable conditions. Several examples are given at the end.
基金Supported in part by the Foundations of the National Natural Science Committee(No.10572154)Jiangxi Province Natural Science Committee(No.0611005)the Foundation of Education Ministry, Jiangxi Province(No.[2006]239), China
文摘A set in Rd is called regular if its Hausdorff dimension coincides with its upper box counting dimension. It is proved that a random graph-directed self-similar set is regular a.e..
基金Supported by NNSF of China and the Foundation of Wuhan University
文摘The structure of any a.s. self-similar set K(w) generated by a class of random elements {gn,wσ} taking values in the space of contractive operators is given and the approximation of K(w) by the fixed points {Pn,wσ} of {gn,ow} is obtained. It is useful to generate the fractal in computer.
基金the National Natural Science Foundation of China
文摘We constructed a class of generalized statistically self-similar set.S and give the necessary and sufficent conditions to ensure a random recursive set being a generalized statistically self-similar set. The statistically self-similar sets defined by Hutchinson,Falconer,Graf are the special cases of ours.
基金partially supported by the foundation of the research item of Strong Department of Engineering Innovation, which is sponsored by the Strong School of Engineering Innovation of Hanshan Normal University, China, 2013partially supported by National Natural Science Foundation of China (No. 11371379)
文摘In this paper, some results on the upper convex densities of self-similar sets at the contracting-similarity fixed points are discussed. Firstly, a characterization of the upper convex densities of self-similar sets at the contracting-similarity fixed points is given. Next, under the strong separation open set condition, the existence of the best shape for the upper convex densities of self-similar sets at the contracting-similarity fixed points is proven. As consequences, an open problem and a conjecture, which were posed by Zhou and Xu, are answered.
文摘In this paper, authors compute the Packing dimension of statistically selfsimilar sets and obtaine the dimension and dimension distribution of statistically self-similar measure.
基金Supported in part by National Natural Science Foundation of China (No.10961003)
文摘In this paper, we present a more simple and much shorter proof for the main result in the paper " An negative answer to a conjecture on the self-similar sets satisfy- ing the open set condition", which was published in the journal Analysis in Theory and Applications in 2009.
基金part by the Foundations of the Jiangxi Natural Science Committee(No:0611005),China.
文摘Falconer[1] used the relationship between upper convex density and upper spherical density to obtain elementary density bounds for s-sets at H S-almost all points of the sets. In this paper, following Falconer[1], we first provide a basic method to estimate the lower bounds of these two classes of set densities for the self-similar s-sets satisfying the open set condition (OSC), and then obtain elementary density bounds for such fractals at all of their points. In addition, we apply the main results to the famous classical fractals and get some new density bounds.
文摘Let d ≥ 1 be an integer and E a self-similar fractal set, which is the attractor of a uniform contracting iterated function system (UIFS) on R^d. Denote by D the Hausdorff dimension, by H^D(E) the Hausdorff measure and by diam(E) the diameter of E. If the UIFS is parametrised by its contracting factor c, while the set ω of fixed points of the UIFS does not depend on c, we will show the existence of a positive constant depending only on ω, such that the Hausdorff dimension is smaller than one and H^O(E) = diam(E)^D if c is smaller than this constant. We apply our result to modified versions of various classical fractals. Moreover, we present a parametrised UIFS, where ω depends on c and show the inequatily H^D(E) 〈 diam(E)^D, if c is small enough.
文摘We study the Hausdorff measure of linear Cantor setE, on the unit interval, under the strong seperated condition. We give a necessary and sufficient condition for ?(E)=∣E∣° by using the contracting ratio and the first gap. This condition is easy to use. Key words linear Cantor set - Hausdorff measure - strong seperated condition CLC number O 174. 12 Foundation item: Supported by the National Natural Science Foundation of China (10171028)Biography: Ma Chao (1975-), male, Ph. D. candidate, research direction: fractal geometry.